On the Moment Generating Function of Pillai's V(P) Criterion in Connection with Three Multivariate Hypotheses in the Complex Case | ||
| The Egyptian Statistical Journal | ||
| Article 3, Volume 31, Issue 2, December 1987, Pages 46-64 | ||
| Document Type: Original Article | ||
| DOI: 10.21608/esju.1987.316580 | ||
| Author | ||
| Nashat B. Saweris | ||
| Abstract | ||
| The moments and moment generating function of Pillai's criterion V(P) received the attention of many authors.In this paper two forms are established for the mgf of V(P) in connection with the following noncentral tests (i) Test of equality of two covariance matrices of two complex normal populations (ii) Testing equality of mean vectors of L-P complex normal populations (MANOVA) and (iii) Test of independence of p and q sets of variates in a (p+q) complex normal populations. It may be noted that V(P) = tr W= ∑ wi, o<w1<…<wp are the latent roots which will arise in each of the underlying tests (i), (ii) and (iii). One of the forms which is established for each of the given tests extends the results obtained by (Khatri(1967) and Pillai (1968)) in the real case. The techniques used for the derivation depend on the properties of complex zonal polynomials, and on some integrals concerning functions of complex argument, which are due to (James (1964) and Khatri (1969)). The derivation of the mgf of V(P) depends on the complex non-central densities associated with the hypotheses (i), (ii) and (iii), which are obtainedby Khatri (1969).However, the lower order moments of V(P) related to the same hypotheses may be verified to follow from the results provided here. | ||
| Keywords | ||
| Complex Gaussian; Non Central Distributions; Moment Generating Functions; Equality of Covariance Matrices; MANOVA; Canonical Correlation | ||
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